Hot dogs and π

I heard that you can calculate the value of π by throwing hot dogs, but I found it hard to believe. I asked the internet.

WikiHow has a full description of how to do it, with photos

I'll summarise, as that's the sort of web site that we came to Gemini to avoid.

Make lines across the floor the same distance apart as the length of a hot dog

Toss the hot dogs among the lines, keeping tallies of the number of throws and the number of times a hot dog ends up crossing a line

Retrieve the hot dogs and rethrow many times to get lots of data

Finally, multiply the throw count by two and divide by the number of line crossers, giving an approximation of π

Someone on WikiHow asked a good question:

> Can I do this during my exam if I don't have a calculator?

But none of this explains why it works. The internet can help us again.

Proper maths explanation of "Buffon's Needle Problem" at wolfram.com

OK, now I'm convinced that it's right, but I'd be happier with a less brain-hurty, more hand-wavy explanation. The best I could come up with was this. If a hot dog lands parallel to the lines, there's no chance that it will cross a line (er, ok, assuming that the lines and the hot dogs are of infinitesimal width). If a hot dog is at right angles to the lines, it's going to cross. In between, it's more or less likely to cross depending on the angle of the hot dog (which is clearly randomly distributed, because of course you threw it with a random twist). If one end of the hot dog is at some arbitrary point, then the probability that it crosses a line depends on how much of the circumference of a circle centred on that point with radius 1 hot dog is on the other side of a line. And circle circumferences involve π. So "all we have to do" is figure the average value of how much of all those possible circles has crossed a line. And if you're not too fussed about the detail, just accept that the number of crossers (c) for a given number of throws (t) is:

...and we can rearrange that for π:

Just remember not to eat the hot dogs after they've been on the floor. Better still, don't eat hot dogs at all because what the hell is in there?

#maths

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